Some routers and switches will drop from gigabit to 100 Mbit if ANY of the devices connected are 100 Mbit. So, take a working gigabit system, plug a Roku into it, and all devices now drop down to 100.
Not true, that is unless the lower speed device is part of the path.
While there may be some switches that drop down to the lowest common denominator, I don't know of any commercial ones.
Wireless yes. Depending on how many radios they have, they can drop down to the speed to the slowest client. Again with some commercial APs they are able to support slow clients and maintain higher speed clients independently.